P?=NP as minimization of degree 4 polynomial, integration or Grassmann number problem, and new graph isomorphism problem approaches

TL;DR

This paper explores novel reformulations of the P vs NP problem using algebra, geometry, Fourier analysis, and continuous optimization, proposing new perspectives and approaches including polynomial discriminants, geometric intersections, Fourier integrals, Grassmann numbers, and graph isomorphism techniques.

Contribution

It introduces multiple innovative reformulations of P vs NP in algebraic, geometric, and analytical frameworks, including Grassmann algebra and new graph isomorphism methods.

Findings

Equivalence of 3-SAT to polynomial zero-finding tested via discriminant.

New geometric formulations for subset-sum and intersection problems.

Grassmann number approach distinguishes strongly regular graphs up to 29 vertices.

Abstract

While the P vs NP problem is mainly approached form the point of view of discrete mathematics, this paper proposes reformulations into the field of abstract algebra, geometry, fourier analysis and of continuous global optimization - which advanced tools might bring new perspectives and approaches for this question. The first one is equivalence of satisfaction of 3-SAT problem with the question of reaching zero of a nonnegative degree 4 multivariate polynomial (sum of squares), what could be tested from the perspective of algebra by using discriminant. It could be also approached as a continuous global optimization problem inside $[0,1]^n$, for example in physical realizations like adiabatic quantum computers. However, the number of local minima usually grows exponentially. Reducing to degree 2 polynomial plus constraints of being in $\{0,1\}^n$, we get geometric formulations as the question if plane or sphere intersects with $\{0,1\}^n$. There will be also presented some non-standard perspectives for the Subset-Sum, like through convergence of a series, or zeroing of $\int_0^{2\pi} \prod_i \cos(\varphi k_i) d\varphi $ fourier-type integral for some natural $k_i$. The last discussed approach is using anti-commuting Grassmann numbers $\theta_i$, making $(A \cdot \textrm{diag}(\theta_i))^n$ nonzero only if $A$ has a Hamilton cycle. Hence, the P$\ne$NP assumption implies exponential growth of matrix representation of Grassmann numbers. There will be also discussed a looking promising algebraic/geometric approach to the graph isomorphism problem -- tested to successfully distinguish strongly regular graphs with up to 29 vertices.